The behaviour of an inviscid, constant density fluid on which no body forces act, may be modelled by the two-dimensional incompressible Euler equations, a non-linear system of partial differential equations. If a fluid whose behaviour is described by these equations, is confined to a space where no fluid flows in or out, the kinetic energy, vorticity integral and enstrophy integral within that space remain constant in time. Solving the Euler equations accompanied by appropriate boundary and initial conditions may be done analytically, but more often than not, no analytical solution is available. The solution, however, can always be approximated and usually a computer is used to provide a numerical approximation.

This thesis aims to introduce mesh refinement into the Mimetic Spectral Element Method (MSEM). The concept of mimetic discretizations is to mimic the properties of continuous Partial Differential Equations (PDEs) discretely. In many discretization methods information is lost in the actual discretization step which is detrimental to the physical fidelity of the approximated solution. Mimetic methods try to prevent this, a feat achieved by combining the fields of differential geometry and algebraic topology. Where differential geometry describes the continuous problem, algebraic topology functions as its discrete equivalent. By accounting for the spatial and temporal geometric objects each physical quantity is associated with, mimetic methods preserve as much as possible of the continuous structure.